Fix a general point \(x \in U\) , and \(\beta \)  the data of a curve class \(\textbf {A}\) , together with compatible contact orders \(p_1,\ldots ,p_k, 0\)  at \(k+1\)  marked points. Let \(\mathcal {M}(X,\beta )_x:= \mathcal {M}(X,\beta ) \times _{X} x\)  be the point constrained moduli space, which forces the marked point \(x_{out}\)  to map to \(x\) . Further define \(\mathfrak {M}^{ev}(\mathcal {X},\beta )_x := \mathfrak {M}^{ev}(\mathcal {X},\beta )\times _X x\) . We equip the natural morphism \(\mathcal {M}(X,\beta )_x \rightarrow \mathfrak {M}^{ev}(\mathcal {X},\beta )_x\)  with a perfect obstruction theory by pulling back the obstruction theory for \(\mathcal {M}(X,\beta ) \rightarrow \mathfrak {M}^{ev}(\mathcal {X},\beta )\) . By [1]} Proposition \(3.19\) , the virtual dimension of this moduli space is \(k-2\) . Now define the following log Gromov Witten invariant:
\(N_{p_1,\ldots , p_k,0}^\textbf {A}\ = \int _{[\mathcal {M}(X,\beta )_x]^{vir}} \psi _{x_{out}}^{k-2}.\) 
